#8772456

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,18,22,25)(19,21,24,30)(20,23,29,32)(26,28,27,31)(33,34)(35,43)(36,37)(38,42)(39,46)(40,41)(44,45)(47,48);(1,2)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(19,24)(21,30)(26,27)(28,31)(33,34,38,42)(35,39,46,43)(36,40,41,37)(44,45,47,48);(2,6)(3,10)(8,12)(11,16)(17,19,22,24)(18,21,25,30)(20,27,29,26)(23,31,32,28)(33,35)(34,39)(36,47)(37,45)(38,46)(40,48)(41,44)(42,43);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20)(18,23)(19,26)(21,28)(22,29)(24,27)(25,32)(30,31)(33,36,38,41)(34,40,42,37)(35,44,46,47)(39,45,43,48);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20)(18,23)(19,26)(21,28)(22,29)(24,27)(25,32)(30,31)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,48)(45,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21)(18,24)(19,25)(20,28)(22,30)(23,27)(26,32)(29,31)(33,37,38,40)(34,41,42,36)(35,45,46,48)(39,44,43,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8679652);